Comparing IR DBI Brane Inflation to Observations
Xingang Chen
陈新刚
CTP, MIT
hep-th/0408084; hep-th/0501184; astro-ph/0507053;
0710.1812, with Rachel Bean, Hiranya Peiris, Jiajun Xu.
Motivation
• Large number of ongoing and forthcoming experiments:
WMAP, SDSS, SNLS, ACBAR, Planck, ACT, Spider, ...
• Specifying inflation model and probing underlying
fundamental theory such as string theory
• Signatures beyond the vanilla LCDM model:
Running of spectral index, Large non-Gaussianities,
Tensor modes, Cosmic strings, …
Approach
• Scan parameter space with minimum requirement:
Enough inflationary e-folds.
• Look for observational signatures in all parameter space
and compare with data.
• Probing string theory through dynamics of our own vacuum
Observational signatures
Specific stringy dynamics
Outline
• Properties of brane inflation: Phase diagrams
• Analytical and numerical properties of IR DBI
• Comparison with data
Brane Inflation in
Warped Compactification
• Brane inflation (Dvali, Tye, 98; Burgess,Majumdar,Nolte,Quevedo,
Rejesh,Zhang;Dvali,Shafi,Solganik,01)
Brane position as inflaton;
Brane annihilation or collision as ending.
• Warped compactification
(Gidding, Kachru, Polchinski, 01;
Klebanov, Strassler, 00; Verlinde, 99;
Randall, Sundrum, 99)
• 6 dimensional bulk
• Warped space generated by
point-like (6d) sources
Phase diagram: UV models
Firouzjahi,Tye,05
(KKLMMT, 03; Silverstein, Tong, Alishahiha,03,04; Shandera,Tye,06 )
• Potential
• Warped space
A-throat
S.R.
S.R.
Slow-roll inflation:
S.R.
S.R.
DBI inflation:
DBI
(Silverstein, Tong, 03)
Geometric Conditions
(Burgess, et.al.,01; X.C,05; X.C.,Sarangi,Tye,Xu,06; Baumann,McAllister,07)
• Planck mass: integration over compact space
• Throats glued to the bulk
: multiplicative factor
from orbifolding
• Maximum separation between branes
: Length scale of A-throat;
: Length scale of bulk
S.R.
S.R.
• Brane inflation is small field:
• Clean separation b.t. Slow-roll and DBI:
DBI
• Slow-roll region: KKLMMT model, 03
Shape of the potential may be adjusted to fit the spectral index;
In the absence of sharp feature,
Non-Gaussianity and running spectral index are unobservable;
Tensor mode is too small to be observed.
(Bean, Shandera, Tye, Xu, 07)
(Berg, Haack, Kors, 04;
Baumann et al, 06;
Burgess,Cline,Dasgupta,Firouzjahi,06;
Krause, Pajer, 07; …)
• DBI region: STA model
(Silverstein, Tong, Alishahiha, 03,04)
Large non-Gaussianity:
Tensor mode:
But inconsistent within GKP-type warped compactification
--- no UV DBI inflation due to probe brane backreactions
(Bean, X.C., Peiris, Xu, 07)
 Antibrane tension cannot drive inflation
So need
 Excessive probe brane backreaction
Requirement:
But:
Note: No comparison with data has been made.
Phase diagram: IR models
(X.C., 04,05; Bean, X.C., Peiris, Xu, 07)
• Potential
,
• Warped space
B-throat
• Multi-throat brane inflation
(X.C. 04)
 Antibrane-flux annihilation (Kachru, Pearson, Verlinde, 01)
 Generate branes as candidate inflatons
 Exit B-throat, roll through bulk, settle down in another throat
 Enough warping: DBI inflation; Flat potential: slow-roll inflation.
S.R.
Slow-roll inflation:
S.R.
DBI
IR DBI inflation:
• For
,
• For
,
DBI
(X.C. 04, 05)
S.R.
DBI
DBI
Geometric conditions are automatically satisfied:
Main Difference Between UV and IR DBI Model
• UV DBI
 Antibrane tension cannot drive inflation,
since it is warped down by the same A-throat warp factor.
An extra, steep, potential is needed to raise the inflationary energy:
with a large m :
• IR DBI
 Speed-limit and antibrane tension are independent of each other:
Speed-limit: B-throat; Inflationary energy: A-throat.
Flexible shape of brane moduli potential:
: over ten orders of magnitude.
Condition for IR DBI inflation:
B-throat warp factor is smaller than
 Flux induced warp factor is exponentially small:
(Giddings,Kachru,Polchinski,01)
Very easy to satisfy the condition.
 Non-trivial condition:
Various back-reactions that chop off the IR end of throat
• Probe brane back-reaction;
(Silverstein,Tong,03; X.C.,04)
Easy to satisfy in IR DBI model.
• Back-reaction from expanding background.
(X.C.,05; X.C.,Tye,06)
Back-reaction from Expanding Background
• From the point of view of closed string creation
Closed string density
(X.C.,05)
Source of the bkgd (N branes)
• From the point of view of open string fluctuations
(X.C., Tye, 06)
Transverse scalar fluctuations on the source branes:
Throat is cut off at
Maximum number of DBI e-folds:
Outline
• Properties of brane inflation: Phase diagrams
• Analytical and numerical properties of IR DBI
• Comparison with data
Brane Dynamics
(X.C.04,05; Bean,X.C.,Peiris,Xu,07)
Two attractor solutions:
• IR DBI inflation:
• Non-relativistic roll, typically fast roll:
(4)
(3)
(2) (1)
1)
: Field theory applies;
2)
: Open string creation
(Stringy quantum fluctuations);
3)
: Closed string creation starts;
4)
: Closed strings smooth out background
(de Sitter back-reaction cuts off the throat).
Density perturbations:
1) Field theory regime
2) Hubble-expansion-induced stringy phase
Density Perturbations
(X.C. 04, 05)
• Field theory regime
 Density perturbations:
 Spectrum index:
• Stringy phase transition:
 Hubble scale < string scale:
 Fluctuation speed < speed of light:
Phase transition at:
if
Estimate the Transition Behavior
(Bean, X.C., Peiris, Xu, 07)
Model: Brane transverse fluctuations:
 Random-walk within the horizon, speed given by H;
 Frozen outside of the horizon.
Field theory regime
Stringy regime
E-fold
Hubble energy
Fluctuation speed
Non-relativistic
World volume
Scalars
Relativistic (superluminal if naïve)
Scalars + strings (branes)
We generalize the behavior of brane transverse fluctuations
relativistically.
Results (in IR DBI region):
 Power spectrum
 Spectral index
 Regional large running
For example,
if
Large non-Gaussianity
• Non-Gaussianities in general single field inflation
are characterized by 5 parameters:
(X.C., Huang, Kachru, Shiu, 06)
c.f. slow-roll inflation, 2 parameters:
(Maldacena, 02; Seery, Lidsey, 05)
• Leading Non-Gaussianities:
Shape:
dependence on the shape of momenta triangle
(Babich, Creminelli, Zaldarriaga, 04)
Running: dependence on the size of momenta triangle
(X.C. 05)
In the absence of sharp features (X.C., Easther, Lim, 06),
running is weak, shape has two categories:
Equilateral shape (DBI inflation)
Local shape (Slow-roll inflation)
• DBI inflation:
(Alishahiha,Silverstein,Tong,04;X.C.,Huang,Kachru,Shiu,06)
• UV DBI inflation (STA model)
• IR DBI inflation
(X.C. 05)
 Different requirements on microscopic parameters.
Geometric conditions have no effect on IR DBI.
 In IR DBI, the large non-G can be small enough to satisfy current bound.
Negative running:
Non-G tends to be the smallest in the entire DBI inflation trajectory.
Small Tensor Mode
• Tensor to scalar ratio:
Lyth Bound:
(Lyth,96; Baumann,Mcallister,06; Lidsey,Huston,07)
is tiny in IR DBI inflation
(Bean, X.C., Peiris, Xu, 07)
Outline
• Properties of brane inflation: Phase diagrams
• Analytical and numerical properties of IR DBI
• Comparison with data
Microscopic Parameters
• Shape of inflaton brane moduli potential:
• Charge of the B-throat:
• Number of inflaton branes:
• Fundamental string scale:
• A-throat warp factor and number of antibranes:
Observables
• Amplitude of power spectrum:
• Scale dependence of power spectrum:
Spectrum index and its running
DBI e-folds
and scale
of the transient large running of
• Non-Gaussianity bound:
• Several consistency conditions, for example:
 Scale – e-fold relation:
 Geometric constraint:
 Number of inflaton branes
Implementing Markov Chain Monte Carlo
Goal: Compare to data directly from microscopic parameters,
using Bayes’ theorem:
: parameters;
: data.
Possible obstacles: Nonlinear and non-transparent relation
between microscopic parameters and observables
Non-Gaussian posterior distributions, curved likelihood surface, etc.
Difficult to search the likelihood surface efficiently
Solution: Reparameterization:
General Procedures
(Bean,X.C.,Hiranya,Xu,07)
1) Extract isolated expression for a small window
in terms of smaller number of parameters
E.g. Full expressions:
have to be solved numerically;
However, approximate expression for observational window:
can be obtained.
Effective parameters:
2) Run a trial MCMC with the effective parameters
,
to ensure that these parameters have simple likelihood surface.
3) Express
(approximately) in terms of microscopic parameters
which provides guidance to the reparameterization
.
E.g.
Using the efold – scale relation:
We approximate:
,
The reparameterization:
These parameters will have simple likelihood surface.
4) Run the full MCMC with
.
Analytical approximation dropped, observables calculated numerically.
5) Transform the likelihood surface of
to the space of the original
parameters
.
Re-weighted to impose any desired priors on
.
The results
Data cannot distinguish
IR DBI from LCDM;
but is able to give interesting
constraints.
Summary of MCMC Results
Microscopic parameters:
• Shape of moduli potential:
Data picks out O(1) value from 10 orders of magnitude that allows IR DBI.
• Fundamental string scale:
Intermediate string scale, intermediate large volume compactification
• B-throat charge:
• Number of inflaton branes:
Flux number
, small number of inflatons is ruled out.
• A-throat minimum warp factor:
A-throat tends to be short; tunneling reheating is possible.
Secondary derived parameters:
• Inflationary phases: the last
non-relativistic fast-roll inflation.
e-folds come from
• The stringy phase transition:
The stringy phase transition happens at the largest scales in the sky;
but its impact extends to shorter scales, generating transient large
running of .
• Inflation scale:
This gives a tiny tensor to scalar ratio:
• Cosmic string tension:
is tension of D-string left over in A-throat after brane annihilation;
F-string tension:
Observational predictions:
• Large, but regional, running of spectral index:
Better theoretical understanding and experimental measurement
may lead to finer structures.
In future experiments, Planck is expected to reach
.
(Planck bluebook)
Reconstructed Power Spectrum
Dashed lines: 1) Single-field slow-roll;
(Peiris, Easther, 06)
2) Empirical power law ansatz.
• Large non-Gaussianities:
In future experiments: on CMB scales, Planck can achieve
;
on LSS scales, high-z galaxy surveys can reach similar or better resolutions.
(Smith, Zaldarriaga, 06; Sefusatti, Komatsu, 07)
Distinguishing IR DBI and other models
• Slow-roll potential with mild features
Usual slow-roll gives negligible running of spectral index:
However, large running of
can be achieved by engineering the potential:
adding mild features, such as periodic ripples. (Bean, X.C., Peiris, Xu, 07)
 Helps to sustain the inflation
 Generating large running of spectral index
varies between
To distinguish, use the non-Gaussianity:
• Non-Bunch-Davies vaccum
(Martin, Brandenberger, 00; ……)
Main difference:
 Non-BD case: new physics energy scale M >> Hubble parameter H,
so field theory apply
 Phase transition in IR DBI: new physics (stringy) scale is
comparable or larger than Hubble parameter H
Generalize slow-roll results (Danielsson, 02; Polarski, Starobinsky, 95)
to case with arbitrary speed of sound (Bean, X.C., Peiris, Xu, 07)
Running spectral index:
 Slow-roll with non-BD: have much smaller
 IR DBI with non-BD: frequent oscillations
, or have frequent oscillations
Conclusions
• Multi-throat brane inflation and IR DBI:
Phase diagram of brane inflation;
Comparision with UV models.
• Warp compactification:
Speed-limit: DBI inflation;
Warped string scale: stringy phase transition.
• Comparing to data:
Current data gives interesting constraints to microscopic parameters.
• Observational predictions:
Regional large running of spectral index; Large non-Gaussianities.
String theory making testable predictions with distinctive signatures;
Probing string theory using cosmological observations.